The Woods-Saxon Potential in the Dirac Equation

The two-component approach to the one-dimensional Dirac equation is applied to the Woods-Saxon potential. The scattering and bound state solutions are derived and the conditions for a transmission resonance (when the transmission coefficient is unity) and supercriticality (when the particle bound state is at E=-m) are then derived. The square potential limit is discussed. The recent result that a finite-range symmetric potential barrier will have a transmission resonance of zero-momentum when the corresponding well supports a half-bound state at E=-m is demonstrated.Comment: 8 pages, 4 figures. Submitted to JPhys

Klein Tunnelling and the Klein Paradox

The Klein paradox is reassessed by considering the properties of a finite square well or barrier in the Dirac equation. It is shown that spontaneous positron emission occurs for a well if the potential is strong enough. The vacuum charge and lifetime of the well are estimated. If the well is wide enough, a seemingly constant current is emitted. These phenomena are transient whereas the tunnelling first calculated by Klein is time-independent. Klein tunnelling is a property of relativistic wave equations, not necessarily connected to particle emission. The Coulomb potential is investigated in this context: it is shown that a heavy nucleus of sufficiently large $Z$ will bind positrons. Correspondingly, it is expected that as $Z$ increases the Coulomb barrier will become increasingly transparent to positrons. This is an example of Klein tunnelling.Comment: 17 page

Low Momentum Scattering in the Dirac Equation

It is shown that the amplitude for reflection of a Dirac particle with arbitrarily low momentum incident on a potential of finite range is -1 and hence the transmission coefficient T=0 in general. If however the potential supports a half-bound state at k=0 this result does not hold. In the case of an asymmetric potential the transmission coefficient T will be non-zero whilst for a symmetric potential T=1.Comment: 12 pages; revised to include additional references; to be published in J Phys

The Continuum Limit and Integral Vacuum Charge

We investigate a commonly used formula which seems to give non-integral vacuum charge in the continuum limit. We show that the limit is subtle and care must be taken to get correct results.Comment: 5 pages. Submitted to JETP Letter

The Strong Levinson Theorem for the Dirac Equation

We consider the Dirac equation in one space dimension in the presence of a symmetric potential well. We connect the scattering phase shifts at E=+m and E=-m to the number of states that have left the positive energy continuum or joined the negative energy continuum respectively as the potential is turned on from zero.Comment: Submitted to Physical Review Letter

Radiation from accelerated perfect or dispersive mirrors following prescribed relativistic asymptotically inertial trajectories

We address the question of radiation emission from both perfect and dispersive mirrors following prescribed relativistic trajectories. The trajectories considered are asymptotically inertial: the mirror starts from rest and eventually reverts to motion at uniform velocity. This enables us to provide a description in terms of in and out states. We calculate exactly the Bogolubov alpha and beta coefficients for a specific form of the trajectory, and stress the analytic properties of the amplitudes and the constraints imposed by unitarity. A formalism for the description of emission of radiation from a dispersive mirror is presented.Comment: 7 figure